MATH 423: Differential Geometry
Spring 2024 Course information
| Class Time/Place: | MoWe 4:00pm–5:15pm, SOND 103 |
| Office: | MP 402 |
| Phone: | 410–455–2458 |
| Email: | rostamian@umbc.edu |
| Office hours: | MoWe 3:00–4:00 |
Textbook and course content
- Textbook
-
John Oprea,
Differential Geometry and Its Applications 2nd edition, American Mathematical Society, 2007.
We will cover the bulk of chapters 1 through 6.
The book is available at the AOK library for online reading and partial downloads. Go to AOK's home page and type “oprea, differential geometry” in the search box. Then in the page that comes up, select the version that has links labeled “PDF Full Text” and “Full Download”.
Personally, I have difficult time grasping mathematics on a computer screen. I much prefer a paper copy.
- Software
-
The main (indeed, the only) point of this course is to
understand the shapes and properties of curves and surfaces
in 3D. That can be done in one's head with some effort, as
it has been done for generations, but that understanding can
be greatly enhanced with the help of graphical capabilities of
the modern computer. In this course we will use Maple
for producing and analyzing graphical objects in 2D and 3D,
as well as performing symbolic calculations and solving
differential equations numerically.
Maple is an all-purpose Computer Algebra System (CAS). Our textbook provides Maple code for many of its examples and illustrations. Learning how to program in Maple is an integral part of this course. I will devote some class time to Maple tutorials. You will need Maple to analyze most of the homework assignments.
Maple Student Edition is priced at $149 per student. I have arranged for a 25% discount for students enrolled in this course, so your cost will be about $112 (plus tax). Ask me for the discount code. The software that you buy this way has no expiration date. (It's possible to buy a 12-month license for $99 but that's not a good investment.) Once you get used to Maple, you will use it all the time in this course and beyond.
Notes
- If you need to make up your mind about staying in this course, you may want to get the free trial version of Maple which will work for 15 days.
-
If you are really strapped for money, you may skip buying Maple
and use the one installed on UMBC's servers. Here is how.
- In a web browser go to https://elum.in/umbcgenlab
- Enter your UMBC username and password. This will set up a Virtual Desktop Environment which will present you with a Microsoft Windows embedded within the browser. (This takes a few minutes.)
- Go to the Start menu within that virtual Windows and search for Maple. Once found, click on "Maple 2023".
- Important! Configure Maple before you begin using it.
- Mathematica is a good alternative to Maple. If you have had a good deal of experience with Mathematica, you may be able to translate our Maple code to Mathematica. But beware that this may not be always easy.
- Prerequisites
- Linear Algebra (Math 221) and
Multivariable Calculus (Math 251)
- Recommended preparation
- CS 201, Math 225, Math 301, and a grade of A in Math 251.
Course Goals/Objectives
Differential geometry has its origins in Karl Friedrich Gauss's paper, Disquisitiones generales circa superficies curvas (General Investigations of Curves Surfaces) which he presented to the Royal Society of Göttingen in 1827. The paper remains quite fresh and readable (in the English translation, anyway) after the passage of almost 200 years. There have been great developments in the subject since then, and it has moved far away from its origins, but in this introductory course we will limit ourselves pretty much to what Gauss did in 1827.
You may download the PDF of the English translation of Gauss's paper from https://www.gutenberg.org/files/36856/36856-pdf.pdf.
The course has the following specific learning goals:
- Master the definitions, theorems, and proofs presented in the book, and in lectures in class.
- Develop a geometric intuition and analytical capability to solve questions posed as examples and exercises.
- Learn how to translate intricate theoretical constructions into static or dynamic computer graphics.
Homework
I will assign homework after every lecture. You are responsible to solving the assigned problems but I will not collect and grade them. Instead, I will have randomly selected students present their solutions to some of the problems in class and will assign grades based on the thoroughness of their presentations. These assessments will amount to 15% of a student's course grade.
Exams
| Homework: | 15% |
| Exam 0: | 5% |
| Exam 1: | 25% |
| Exam 2: | 25% |
| Final Exam: | 30% |
We will have one take-home exam based on the course's prerequisite materials at the beginning of the semester, two exams during the semester, and a final exam at the end. The overall course grade will be be determined according to the weights attached to the various components as shown in the adjacent table. Letter grades will be determined according to:
if { grade ≥ 85: A}
else if { grade ≥ 75: B}
else if { grade ≥ 65: C}
else if { grade ≥ 55: D}
else F
I will make and grade the exams in a fair and reasonable way, but sorry, no "curving" in this course.
Cool demos (made in Maple)
Course calendar and activity log
| Calendar | |
|---|---|
| Mon Jan 29 | Exam 0 on the prerequisites |
| Wed Jan 31 | Can you do this animated plot of the sine function in Maple? |
| Mon Feb 5 | |
| Wed Feb 7 | Homework #1: hw01.pdf |
| Mon Feb 12 | Homework #2: hw02/ |
| Wed Feb 14 | |
| Mon Feb 19 | Homework #3: hw03/ |
| Wed Feb 21 | |
| Mon Feb 26 | Exam1 due on March 4 |
| Wed Feb 28 | No homework assigned today |
| Mon Mar 4 | |
| Wed Mar 6 | Homework #4: hw04/ |
| Mon Mar 11 | |
| Wed Mar 13 | |
| Mon Mar 18 | Spring Break; no class today |
| Wed Mar 20 | Spring Break; no class today |
| Mon Mar 25 | Notes of differential geometry |
| Wed Mar 27 | Exam 2 due on Wednesday April 3 |
| Mon Apr 1 | |
| Wed Apr 3 | Homework #5: hw05/ |
| Mon Apr 8 | |
| Wed Apr 10 | The Physics of soap films and bubbles |
| Mon Apr 15 | |
| Wed Apr 17 | Homework #6: hw06/ |
| Mon Apr 22 | Homework #7: hw07/ |
| Wed Apr 24 | |
| Mon Apr 29 | |
| Wed May 1 | |
| Mon May 6 | |
| Wed May 8 | |
| Mon May 13 | Final Exam |
Miscellaneous notes & comments
Registrar's info
Registrar's Office Dates and Deadlines
Configuring Maple
Read this
before you begin using Maple
Maple basics
maple-basics-1.mw
maple-basics-2.mw
christoffel-and-geodesics.mw
bonet-fundamental-theorem.pdf
gauss-bonnet.pdf
1827-Gauss-Latin.pdf
Exam 0
exam0-solutions.pdf
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